The moderation effect of social capital in the relationship between own income, social comparisons and subjective well-being: Evidence from four international datasets

In this paper we check whether social capital changes the association of subjective well-being with own income and social comparisons. We use panel data from Germany and publicly available data from three international surveys, for a total of nearly 500,000 respondents from industrial countries. Results show that the association of own income and social comparisons to subjective well-being weakens for individuals with high social capital. This finding holds in a variety of settings, and is robust to various measures of subjective well-being, of social capital, and of social comparisons. We also find evidence indicating that the role of social capital is, at least in part, causal. Finally, our findings support the macro-level implication that income differences are less related to subjective well-being differences in countries with high social capital.


S1 Appendix. Online Appendix
Note: The high collinearity on the income and reference income should not cause any concern, as it is a mechanical consequence of the construction of the reference income variable.Table A13: Robustness check of regional level analysis on EU-SILC (2013) data using the 90/10 ratio as a measure of income inequality. (1) (2) (3) Share of people with SC = 2 -0.521 * * * -0.641Standard errors in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001 Note: The unit of analysis are regions.All coefficients are standardised for comparability.Data for Gini, 90/10 ratio and GDP are from Eurostat.Share refers to share of national equivalised income and cut-off refers to the top cut-off point.All variables are standardised for comparability.

Size effects
Well-being differences between rich and poor people form micro-results and macro results: the aim is to relate micro-to macro-results by comparing the size of estimated effects of the subjective well-being differences between rich and poor people (corrected for by dividing this difference by the average life satisfaction).We are interested in understanding what is the difference in life sfatisfaction between income groups (top and bottom) when people have high social capital, and the same difference when people have no social capital.The expectation is that this difference would be higher in people who have no social capital.We are intterested in the following reduced computation: In terms of making these results comparable to between micro and macro estimations we do the following: • We compute the average levels income of the top and bottom income quintiles and multiply them by the coefficient of the interaction effect of social capital (=2) and income, to get the the effect at the mean: −0.24 * 7.87 = −1.89and −0.24 * 2.97 = −0.72 • similarly, we compute the average reference income of people in the top and bottom income quintiles, and multiply it by the interaction coefficient of the social capital (=2) and reference income (0.148 * 7.50 = 1.11 and 0.148 * 3.36 = 0.49) • So the life satisfaction gap between top and bottom income quintiles with high social capital, minus the life satisfaction gap of rich and poor people having no social capital, corrected of the average levels of life satisfaction to obtain relative differences is equal to ((−1.89 + 1.11) + (−0.7 + 0.49))/averageLif eSatisf action , which is almost −0.145.This suggests that the life satisfaction gap between top and bottom income quintiles would be reduced by 0.145 for people with high levels of social capital, compared to well-being differences of people with no social capital.
• At the macro level, we run the unstandardized regression of life satisfaction gap between the top and bottom income quintiles divided by average life satisfaction on the share of people with high social capital, Gini index and GDP, and multiply the coefficient on the share of high social capital people (−0.354) with the difference in shares of high social capital people in countries at the maximum level and the minimum level in the sample, namely: (0.774 − 0.198) * −0.354 = −0.0204(as seen in table A8, second row).This suggests that increasing the share of people in a country will reduce the life satisfaction gap between the top and bottom income quintiles.

Integrated European Values Study -World Values Study
Table A23: OLS with robust standard errors using WVS-EVS (waves 3-6) data: detailed results.
(    4 German SOEP Note: * p < 0.05, * * p < 0.01, * * * p < 0.001.Omitted categories: "Social capital index = 0 * log of individual income", "Social capital index = 0 * log of reference income".Controls: regional dummies, year dummies.We note that the main effect of social capital becomes negative when the estimation includes the interaction term between reference income and social capital.We do not have an explanation for that, hence we suggest further research on this topic. Table A33: Robustness check using the single dummies for social capital rather than the index (SOEP). (

Lewbel Method of heteroskedasticity generated instruments
There are various reasons to believe that social capital is endogenous to wellbeing, as the association between social capital, social comparisons and subjective wellbeing may be driven by omitted variables or reverse causality.We account for endogeneity using a Two-Stages Least Squares (2SLS) instrumental variable approach.Specifically, we instrument the main effect of social capital, and its interaction terms with absolute and reference income.Identifying a proper instrument for social capital is difficult, as most of the factors affecting people's social life will likely affect their wellbeing as well.To overcome this problem we use the method of generated instruments proposed by Lewbel (2012).This approach allows to identify a causal model without imposing the exclusion restriction which is typically required in a standard 2SLS, while instead exploiting the heteroskedasticity of the first step equation to construct the instruments (Lewbel, 2012).As discussed by Lewbel (2012) and Baum and Lewbel (2019), this method is valid when the endogeneity of the instrumented variable comes from an error component that appears in both the reduced form and structural equations.In the case of social capital, the error component may be unobserved individual characteristics, such as personality traits, which affect both subjective wellbeing and social capital.
Formally, we implement the two-stage estimator proposed by Lewbel in the following way: to begin, we regress each endogenous variable on all of the control variables from our main equation of subjective wellbeing X, and the vector of residuals µ i are retrieved.More specifically, we run the following first-stage regressions: If the residuals from Eq. 2, 3, 4 are heteroskedastic, instruments can be generated by multiplying them with each of the mean-centered observed variables (X j ), as follows: where j corresponds to a given control variable from vector X, and μ are the vectors of residuals from Eq. 2, 3 and 4. Hence, for each endogenous variable the number of generated instruments Z is equal to the number of control variables included in the vector X.The predicted values are then used in the second step of the 2SLS framework as follows: The Lewbel approach relies on the same assumptions of a standard instrumental variable model, with the addition of two extra conditions.The first is that there exists heteroskedasticity in the first stage equation, that is Cov(Z, µ 2 ) ̸ = 0, where Z is the vector of instruments constructed from some or all of the variables included in the vector of controls of the structural equation X.The second condition is that there exists a Z ⊆ X for which Cov(Z, µϵ) = 0, where ϵ is the error term of the structural equation of wellbeing, which would allow the constructed instruments to satisfy the exclusion restriction.
The intuition behind the mechanics of the Lewbel approach comes from a standard linear regression mechanics: the residuals are by construction exogenous to the right hand side variables if the model is correctly specified.This means that if the structural form is correctly specified, the remaining errors are idiosyncratic (Lewbel 2012).Hence, if the chosen X are exogenous in the structural equation, the instruments created on those X are also exogenous, and will affect the outcome variable only via the endogenous regressor.As Lewbel and Baum and Lewbel discuss, if this assumption does not hold, that is if Cov(Z, µϵ) ̸ = 0, bounds on the causal parameters can still be obtained as long as this covariance is not too large (Lewbel, 2012;Baum and Lewbel, 2019).Lastly, If the residuals are heteroskedastic, they contain information about the the variation of the outcome (endogneous) variable, which makes the instruments relevant.
A plausible cause of heteroskedasticity in equations 2, 3 and 4 may come from the non constant variance in the distribution of the residuals of social capital over the age distribution.We test this with a Breush-Pagan test, which confirms this hypothesis with p-values consistently smaller than 0.001.Additionally, age is exogenously determined with respect to subjective wellbeing, which makes it a valid source to construct the instrument on.Hence, in short, we construct the instruments Z on demeaned age and age squared, multiplied by the residuals of equations 2, 3 and 4.  Instrumented variables are the "social capital", "social capital * income" and "social capital * reference income ".The method is 2SLS with robust standard errors, where the employed instruments have been generated using the Lewbel method.The social capital variable is treated as continuous to limit the number of instruments necessary for identification.Controls included in each of the estimated equations are the same those included in the main OLS results.

Error Propagation method
We estimate the errors of our moderation effects using the error propagation method.
Formally, the formula for the error propagation method is defined as follows: let f (x 1 , x 2 , ..., x n ) be a function which depends on n variables x 1 , ..., x n and the the uncertainty around each variable be defined as x i ± ∆x i , where ∆x i is the error.
If the variables are correlated, the function error ∆f is calculated as follows where C i,k is the covariance between the couples of variables, C i,k = cov(x i , x k ).
In our case, the function is the moderation effect, which is defined as the ratio between the estimated coefficients on the interaction term of social capital with income (reference or absolute), and income.In particular: f = SC * Income Income where income is either absolute income or reference income, and both terms in the ratio are the estimated coefficients from the equation of well-being on social capital, income and reference income and their interaction (Table 1 in the main document), which we assume are correlated.
After some computation the formula to obtain the errors on the moderation effect can be written as follows: where se stands for standard error of the incomes and interaction coefficients, and the rest are all estimated coefficients.C SC * Inc,Inc is the covariance between estimated coefficients.

Table A2 :
The role of social capital in the relationship between absolute income and social comparisons with subjective wellbeing.Regression results from four datasets.

Table A6 :
Descriptive statistics (EU-SILC, 2013): micro-data p < 0.05, * * p < 0.01, * * * p < 0.001 Note: The unit of analysis are countries.All coefficients are standardised for comparability.Data for Gini, 50/10 ratio and GDP are from Eurostat.Share refers to share of national equivalised income and cut-off refers to the top cut-off point.All variables are standardised for comparability. *

Table A12 :
Robustness check of country level analysis on EU-SILC (2013) data using the 90/10 ratio as a measure of income inequality.Note: The unit of analysis are regions.All coefficients are standardised for comparability.Data for Gini, 90/10 ratio and GDP are from Eurostat.Share refers to share of national equivalised income and cut-off refers to the top cut-off point.All variables are standardised for comparability.

Table A19 :
Detailed results of the country level analysis using ESS (2018) data.

Table A20 :
Descriptive statistics: ESS (2018), country level data.Across European countries, the life satisfaction gap between rich and poor people negatively correlates with social capital (ESS, 2018).
Note: Social capital is measured as the share of respondents with a social capital index = 2.

Table A21 :
Robustness check of country level analysis on ESS (2018) data using the 90/10 ratio as a measure of income inequality.
* p < 0.05, * * p < 0.01, * * * p < 0.001 Note: The unit of analysis are countries.Data for Gini and 90/10 ratios are from Eurostat, data for GDP are from the World Bank.Share refers to share of national equivalised income and cut-off refers to the top cut-off point.All variables are standardised for comparability.

Table A22 :
Robustness check of country level analysis on ESS (2018) data using the 50/10 ratio as a measure of income inequality.

Table A28 :
Detailed results of the country level analysis using WVS-EVS (waves 3-6) data.p < 0.05, * * p < 0.01, * * * p < 0.001 Note: OLS with robust standard errors.The unit of analysis are countries.All variables are standardised for comparability. *

Table A30 :
List of developed countries included in the analysis of WVS-EVS (waves 3-6) data.

Table A31 :
Robustness check of country level analysis on WVS data using the 90/10 ratio as a measure of income inequality p < 0.05, * * p < 0.01, * * * p < 0.001 Note: The unit of analysis are countries.Data for Gini, 90/10 and for GDP are from the World Bank.Share refers to share of national equivalised income and cut-off refers to the top cut-off point.All variables are standardised for comparability. *

Table A32 :
OLS with robust standard errors and individual fixed effects using SOEP data: detailed results.
OLS with individual fixed effects and robust standard errors.Dependent variable: Life satisfaction (0-10).Controls: sex (omitted due to fixed effects), age, age squared, marital status, years of education, labour market status, house owner, disability status of individual, living in East Germany, regional dummies, year dummies.p < 0.05, * * p < 0.01, * * * p < 0.001 Standard errors in parentheses. *

Table A34 :
Correlations tableSocial capital index Soc Gath Monthly Help Fre Monthly Volunt Monthly Local participation Individual income Reference income